How is the Average Force Calculated for a Superball Bouncing Between Two Walls?

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In summary, the conversation discusses a problem from Kleppner and Kolenkow involving a "superball" bouncing between two surfaces with neglect to gravity and perfectly elastic collisions. The main points of the conversation revolve around finding the average force on each wall and how the bounce rate is affected when one of the surfaces is moved towards the other. The solution provided in the pdf file uses variable names V for wall speed and v for ball speed, and notes that the ball speed increases by 2 V with each bounce cycle. There is a discrepancy with the initial velocity of the ball before impact, but it is resolved by understanding the variable names used in the problem.
  • #1
WiFO215
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1
NOTE: I AM NOT POSTING A HW QUESTION.

It is the last problem, 4.29.

4.29 A "superball" of mass m bounces back and forth between two surfaces with speed Vo. Gravity is neglected a nd the collisions are perfectly elastic.
a. Find the average force F on each wall.
b. If one surface is slowly moved toward the other with speed V «v,
the bounce rate will increase due to the shorter distance between colli-
sions, and because the ball's speed increases when it bounces from the
moving surface. Find F in terms of the separation of the surfaces, x.
(Hint: Find the average rate at which the ball's speed increases as the
surface moves.)

Here is the link to the answer : http://physics141.uchicago.edu/2002/hw4.pdf

This is a problem from Kleppner and Kolenkow. I have a problem with the working. It is claimed that after every collision, the velocity changes by 2V.
My point is that initial velocity before striking the wall was Vo towards the left and after the collision, the velocity is Vo + 2V towards the right, thus making the change of velocity 2(Vo + V) and not just 2V. Can someone justify how it is that one can solve the problem as done above?
 
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  • #2
For the moving wall, if the ball speed is Vb and the wall speed is Vw then the collision speed is (Vb + Vw) and the rebound speed is - (Vb + 2 Vw) (using a fixed frame of reference).

In the pdf file, V is used for the wall speed, and v is used for the ball speed, and it's noted that with each bounce cycle, the ball speed increases by 2 V, which seems to be the same as what you're asking. The distance between the walls is defined as x = L - Vt. (I swapped uppercase L for the lower case l in the pdf file). I think once you understand the variable names, then the rest of the problem solution makes sense.
 
Last edited:
  • #3
Jeff Reid said:
For the moving wall, if the ball speed is Vb and the wall speed is Vw then the collision speed is (Vb + Vw) and the rebound speed is - (Vb + 2 Vw) (using a fixed frame of reference).

You mean velocity. Collision velocity, i.e. velocity of ball before impact, is only Vb, not Vb + Vw.
 

1. How does a Superball bounce between two walls?

A Superball bounces between two walls due to the principles of physics, specifically the conservation of energy and momentum. When the ball hits the first wall, it compresses and stores energy. As it rebounds off the wall, this stored energy is released, propelling the ball towards the second wall. The process repeats until the energy is dissipated through friction and the ball eventually stops bouncing.

2. Can the height of the walls affect the behavior of a Superball?

Yes, the height of the walls can affect how a Superball bounces. The higher the walls, the longer the ball has to accelerate towards the ground and gain energy. This can result in a higher bounce compared to lower walls.

3. Why does a Superball bounce higher than other types of balls?

A Superball is made from a very dense rubber material that is highly elastic. When the ball hits a hard surface, such as a wall, it is able to quickly compress and store energy. This stored energy is then released as the ball rebounds, resulting in a higher bounce compared to other types of balls made from less elastic materials.

4. Is the angle of the ball's trajectory important in a Superball between two walls?

Yes, the angle of the ball's trajectory is important in determining how a Superball will bounce between two walls. If the ball hits the walls at a perpendicular angle, it will have a more predictable and consistent bounce compared to hitting the walls at an angle. The angle of the walls can also affect the ball's trajectory and bounce.

5. Can the surface of the walls affect the behavior of a Superball?

Yes, the surface of the walls can affect how a Superball bounces. If the walls are smooth and hard, the ball will have a more predictable and consistent bounce compared to walls with a rough or soft surface. This is because a smooth surface reduces the amount of energy lost through friction, allowing the ball to maintain its bounce for longer.

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